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Concept
Intersection (set theory)
Summary
In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
The intersection of more than two sets (generalized intersection) can be written as:
which is similar to capital-sigma notation.
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
The intersection of two sets and denoted by , is the set of all objects that are members of both the sets and
In symbols:
That is, is an element of the intersection if and only if is both an element of and an element of
For example:
The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
The number 9 is in the intersection of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.
We say that if there exists some that is an element of both and in which case we also say that . Equivalently, intersects if their intersection is an , meaning that there exists some such that
We say that if does not intersect In plain language, they have no elements in common. and are disjoint if their intersection is empty, denoted
For example, the sets and are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.
List of set identities and relations and Algebra of sets
Binary intersection is an associative operation; that is, for any sets and one has
Thus the parentheses may be omitted without ambiguity: either of the above can be written as . Intersection is also commutative. That is, for any and one has
The intersection of any set with the empty set results in the empty set; that is, that for any set ,
Also, the intersection operation is idempotent; that is, any set satisfies that . All these properties follow from analogous facts about logical conjunction.
Intersection distributes over union and union distributes over intersection.
Official source
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